Fractions and algebraic reasoning

1

Fractions and algebraic reasoning

1_{Education and Learning, Dalarna University and Department of Mathematics and Science}

Education, Stockholm University, Sweden; hei@du.se

2 _{Department of Mathematics and Science Education, Stockholm University, Sweden;}

lovisa.sumpter@mnd.su.se

Traditional, algebra is seen as a tool for working with generalities after the students have developed an understanding of fractions (Lee & Hackenberg, 2014). On the other hand, earlier research show that children can deal with real numbers, fractions included, using algebra even before assimilating the concept of fractions (see Davydov, 2008/1986; Radford, 2013). The data used in this paper was produced in collaboration with teachers and students in a Swedish grade 4, where most of the students were Swedish as second language learners. The data consists of one videotaped lesson. The task used in the lesson, measuring and comparing different lengths, was designed with inspiration from Davydov (2008/1986). The problem the students faced was to discuss results representing by an integer and a fraction. In this presentation we will discuss what arguments used to develop a model for fractions through collective mathematical reasoning. According to a pretest the students were not familiar with either algebra or fractions. The question posed for this study is “What are the mathematical properties used in arguments in the collective reasoning during this problem solving?”. The analysis of the data was conducted using analytical tool suggested by Sumpter and Hedefalk (2015) focusing on mathematical properties in the chain of arguments. A preliminary result indicates that the arguments given in this situation of collective reasoning were both based on mathematical properties in relation to algebra and in relation to fractions. More specifically, there was an inter-play between algebraic thinking (c.f. Radford, 2014) and reasoning based on fractional knowledge (c.f. Lee & Hackenberg, 2014) when the students found relationships between a quantity as a whole and some parts of this whole (c.f. Davydov, 2008/1986) .

References:

Davydov, V. V. (2008/1986). Problems of Developmental Instruction. A theoretical and

experimental psychological study. New York: Nova Science Publishers, Inc.

Lee, M., & Hackenberg, A. (2014). Relationships between fractional knowledge and algebraic reasoning: The case of Willa. International Journal of Science and Mathematics Education, 12: 975-1000.

Radford, L. (2014). The Progressive Development of Early Embodied Algebraic Thinking.

Mathematics Education Research Journal, 257-277.

Sumpter, L., & Hedefalk, M. (2015). Preeschool children's collective mathematical reasoning during free outdoor play. The Journal of Mathematical Behavior, (30) 1-10.